Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401790 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2485 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14823 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,16,25,17,8,15,24)(4,18,20,27,21,28,22,9)(14,26,19,29,31,32,30,23);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,24)(11,13)(12,25)(14,20)(18,23)(21,30)(22,31)(27,32)(28,29);(1,4)(2,9)(3,14)(5,22)(6,20)(7,21)(8,23)(10,18)(11,27)(12,28)(13,29)(15,19)(16,30)(17,31)(24,32)(25,26);(2,10,11,12)(8,24,13,25)(9,18,27,28)(23,32,29,26)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318867, #72318910, #72318963, … |
| maximal normal subgroup of | #72318867, #72318910, #72318963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308986);